It's well known that if is a a Pythagorean triple, that is, if is a solution in positive integers to the 90 degree triangle equation , then 3 and 4 each divides or , and 5 divides or where, of course, is the smallest such solution.
A 120 degree triple, , is a solution in positive integers to the 120 degree triangle equation
So, naturally, one wonders if a similar relationship exists between the positive integer solutions of 120 degree triangles and the smallest such solution, ? It does, almost. To find such a relationship it's necessary to include an term.
All primitive solutions to a 120 degree triple , are given by the parametric equations:
If and are considered the same solution, then the first 6 primitive solutions in order of smallest value for c are,
Notice that, in each case, 3 and 5, each, divides one of , or , and 7 divides one of , , or .
From (37),
A 60 degree triple, , is a solution in positive integers to the 60 degree triangle equation
Note that
Hence, if is a 120 degree triple then and are 60 degree triples.
Here is a ``neat'' way to construct these three triangles.
On line layout line segments and having lengths and respectively, where and are the side lengths of a 120 degree triangle. On and below construct equilateral triangle with sides of length . On and above construct equilateral triangle with sides of length . Hence and are each 60 degrees. So point lies on line segment and is 120 degrees. Draw line segment . Thus, the construction shows the 120 degree triangle and its two associated 60 degree triangles and .
Let . If , and are positive integers, then is a 60 degree triple. If, additionally, , and are pairwise relatively prime, then is a primitive 60 degree triple. The first seven such triples in order of the smallest value for are,
Notice that, in each case, 3 and 5, each, divides one of , or , and 7 divides one of , , or .
Hence is a 120 degree triple. So, from claim (8),
The drawing below shows two 60 degree triangles
and
along with their associated 120 degree triangle
.
To find relative prime non-zero integer solutions to , set
where
And where and are positive integers, , and is not a multiple of Then .
Note that if integers and have the same sign then and have the same divisors. And if and have opposite parity then and have the same divisors. So to change table (6) to a table of divisors of 60 degree triples all that's necessary is to change to